Summary: 1
WHEN DO THREE LONGEST PATHS HAVE A COMMON VERTEX?
MARIA AXENOVICH
Abstract. It is well known that any two longest paths in a connected graph share a vertex. It is also known
that there are connected graphs where 7 longest paths do not share a common vertex. It was conjectured that
any three longest paths in a connected graph have a vertex in common. In this note we prove the conjecture
for outerplanar graphs and give sufficient conditions for the conjecture to hold in general.
1. Introduction
Gallai asked in 1966 whether every connected graph has a vertex that appears in all its longest paths. Zam-
firescu [6] found a graph with 12 vertices in which there is no common vertex to all longest paths. See Voss, [5]
for related problems. It is well-known that every two longest paths in a connected graph have a common vertex.
Skupien [4] obtained, for k 7, a connected graph in which some k longest paths have no common vertex, but
every k - 1 longest paths have a common vertex. Klavzar et al. [3] showed that in a connected graph G, all
longest paths have a vertex in common if and only if for every block B of G all longest paths in G which use at
least one edge of B have a vertex in common. Thus, if every block of a graph G is Hamilton-connected, almost
Hamilton-connected, or a cycle, then all longest paths in G have a vertex in common. It was also proved in [3]
that in a split graph all longest paths intersect. Balister et al. [1] showed that all longest path of a circular arc
graphs share a common vertex. Still, the following conjecture remains open in general:
Conjecture 1 For any three longest paths in a connected graph, there is a vertex which belongs to all three of
them.